The line that touches two wheels at once, and why a belt needs it
A drive belt looping around two pulleys rides along a line that grazes both wheels without cutting into either, a single straight line tangent to two circles at the same time. That line is a common tangent, and how many of them exist tells you instantly how the two circles are arranged.
What Are Common Tangents?
A common tangent to two circles is a straight line that is tangent to both circles at once, touching each at exactly one point without crossing it. Depending on how the two circles are positioned, they may share several such lines, one, or none at all.
Common tangents come in two kinds, and the difference is which side of the line the circles sit on. Each individual touch still obeys the basic rule that a tangent meets the radius at $90°$, so common tangents build directly on the geometry of single circles.
By the end you will be able to count the common tangents of any two circles and compute their lengths.
What Is the Difference Between Direct and Transverse Common Tangents?
The split is about which side of the tangent line the two circles lie on.
Direct (external) common tangent: both circles lie on the same side of the line. These are the "outer" tangents; a belt running around two pulleys the ordinary way follows the direct tangents.
Transverse (internal) common tangent: the circles lie on opposite sides of the line, which therefore passes between them and crosses the line of centres. A crossed belt (a figure-eight drive) follows the transverse tangents.
A neat property pins down where each type crosses the line of centres: the direct common tangents meet the line of centres at a point that divides it externally in the ratio $r_1 : r_2$, while the transverse common tangents meet it at a point that divides it internally in the same ratio. The centres $O_1$ and $O_2$ are found as the center of each circle.
How Many Common Tangents Can Two Circles Have?
How many common tangents can be drawn to two circles? The answer depends entirely on the distance $d$ between the centres compared with the radii $r_1$ and $r_2$. Here is every case.
Configuration | Condition | Direct | Transverse | Total |
|---|---|---|---|---|
Circles fully apart | $d > r_1 + r_2$ | 2 | 2 | 4 |
Touching externally | $d = r_1 + r_2$ | 2 | 1 | 3 |
Intersecting at two points | $\lvert r_1 - r_2\rvert < d < r_1 + r_2$ | 2 | 0 | 2 |
Touching internally | $d = \lvert r_1 - r_2\rvert$ | 1 | 0 | 1 |
One inside the other | $d < \lvert r_1 - r_2\rvert$ | 0 | 0 | 0 |
The count drops as the circles move closer: four when separate, then three, two, one, and finally zero once one circle sits fully inside the other. The two touching cases are worked out in detail under circles touching each other. When $d = 0$ the circles are concentric circles, and no common tangent exists.
How Do You Find the Length of a Common Tangent?
The length of a common tangent segment (the part between the two contact points) has a clean formula for each type, and both come from dropping the problem into a right triangle.
Direct common tangent length:
$$L_{\text{direct}} = \sqrt{d^2 - (r_1 - r_2)^2}$$
Transverse common tangent length:
$$L_{\text{transverse}} = \sqrt{d^2 - (r_1 + r_2)^2}$$
Here $d$ is the distance between the centres, and $r_1$, $r_2$ are the radii. In each case you slide one radius parallel to itself to build a right triangle whose hypotenuse is $d$ and whose legs are the tangent length and the radius gap (difference for direct, sum for transverse). The transverse formula only gives a real length when $d > r_1 + r_2$, which matches the table: transverse tangents exist only when the circles are fully apart.
Examples of Common Tangents
Example 1
Two circles of radii $5$ cm and $2$ cm have centres $10$ cm apart. How many common tangents do they share?
Compare $d$ with $r_1 + r_2 = 5 + 2 = 7$. Since $d = 10 > 7$, the circles are fully apart, so they share four common tangents.
Example 2
Two circles of radii $4$ cm and $3$ cm have centres $7$ cm apart. A student uses the transverse formula $\sqrt{d^2 - (r_1 + r_2)^2}$ and gets $\sqrt{49 - 49} = 0$, then reports "no transverse tangent length." What is really going on?
Reaching for the transverse formula first hides the arrangement. Check the configuration before choosing a formula: here $d = 7 = r_1 + r_2$, so the circles touch externally, and the touching case has no transverse tangent at all, which is exactly why the formula collapses to $0$.
There are still three common tangents in this case, two direct and one at the contact point. The direct tangent length is:
$$L_{\text{direct}} = \sqrt{d^2 - (r_1 - r_2)^2} = \sqrt{49 - 1} = \sqrt{48} = 4\sqrt{3} \text{ cm}$$
Classify the circles first, then apply the matching formula.
Example 3
Find the length of the direct common tangent of two circles with radii $6$ cm and $2$ cm whose centres are $5$ cm apart.
$$L_{\text{direct}} = \sqrt{d^2 - (r_1 - r_2)^2} = \sqrt{5^2 - (6-2)^2} = \sqrt{25 - 16} = \sqrt{9} = 3 \text{ cm}$$
The direct common tangent is $3$ cm long.
Example 4
Find the length of the transverse common tangent of two circles with radii $4$ cm and $1$ cm whose centres are $13$ cm apart.
$$L_{\text{transverse}} = \sqrt{d^2 - (r_1 + r_2)^2} = \sqrt{13^2 - (4+1)^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ cm}$$
The transverse common tangent is $12$ cm long.
Example 5
Two circles intersect at two distinct points. How many common tangents do they have?
Intersecting circles satisfy $|r_1 - r_2| < d < r_1 + r_2$, which gives two common tangents, both direct. The transverse tangents disappear the moment the circles overlap.
Example 6
A circle of radius $3$ cm lies entirely inside a circle of radius $8$ cm, with centres $2$ cm apart. How many common tangents are there?
Compare $d$ with $|r_1 - r_2| = |8 - 3| = 5$. Since $d = 2 < 5$, the small circle sits fully inside the large one without touching, so there are zero common tangents. Each tangent from an outside point, by contrast, obeys the equal-length rule of tangents from an external point.
Where Do Common Tangents Matter?
Common tangents are the geometry of anything that wraps two round parts.
Belt and chain drives. An open belt follows the direct common tangents of two pulleys; a crossed belt follows the transverse ones and reverses the driven wheel's spin.
Roller and cam systems. A flat bar resting on two rollers sits along a direct common tangent, so its height is set by the tangent geometry.
Robotics and path planning. Routing a cable or a robot arm around two circular obstacles uses common-tangent lines as the shortest grazing paths.
The pulley case is the clean real-world anchor: belt length calculations start from the direct-tangent formula above, a standard result in the geometry of tangent lines to circles. Choose the wrong tangent type and the belt runs the wrong way. These relationships also feed into the other standard circle theorems.
Mistakes to Watch For
Mistake 1: Swapping the direct and transverse formulas
Where it slips in: Picking a formula before checking which tangent type the problem wants.
Don't do this: Use $d^2 - (r_1 + r_2)^2$ for a direct tangent, or $d^2 - (r_1 - r_2)^2$ for a transverse one.
The correct way: Direct uses the difference of radii, transverse uses the sum. A memory hook: direct tangents keep the circles on one side, so the radii nearly cancel (difference); transverse tangents pass between, so the radii pile up (sum). The rusher who grabs either formula without labelling the type gets a wrong length or a negative under the root.
Mistake 2: Applying a length formula before classifying the circles
Where it slips in: Circles that touch or overlap, where some tangents do not exist.
Don't do this: Plug into the transverse formula for touching or intersecting circles and trust the number.
The correct way: First compare $d$ with $r_1 + r_2$ and $|r_1 - r_2|$ to fix the configuration and the tangent count. Only then apply the formula for the tangents that actually exist. A zero or imaginary result is the geometry telling you that tangent type is absent.
Mistake 3: Forgetting internal circles have no common tangent
Where it slips in: One circle nested inside another.
Don't do this: Assume every pair of circles shares at least one tangent.
The correct way: When $d < |r_1 - r_2|$, the inner circle is fully enclosed and there are zero common tangents. The second-guesser who keeps searching for a tangent line is looking for something that cannot exist.
Conclusion
A common tangent touches two circles at once, and two circles share between zero and four of them.
Direct tangents keep both circles on the same side; transverse tangents pass between the circles.
The count is fixed by the centre-distance: 4 apart, 3 touching externally, 2 intersecting, 1 touching internally, 0 nested.
Direct length is $\sqrt{d^2 - (r_1 - r_2)^2}$; transverse length is $\sqrt{d^2 - (r_1 + r_2)^2}$.
The most common mistake is swapping the sum and difference of radii between the two length formulas.
To work common-tangent problems through with a teacher, explore Bhanzu's geometry tutor, our high school math tutor sessions, or ongoing math tutoring. To watch a trainer count and measure common tangents live, you can book a free demo class.
Read More
Tangents and normals — how tangent lines behave at a single circle.
Equation of a circle — the coordinate form used to locate two circles before finding their tangents.
Circles centered at the origin — the simplest coordinate circle to test tangent lines against.
Equal and unequal chords — how chord lengths relate to their distance from the centre.
Parts of a circle — the centre, radius, and chord a tangent construction relies on.
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